Approximation algorithms for the graph balancing problem with two speeds and two job lengths

Consider a set of n jobs and m uniform parallel machines, where each job has a length $$p_j \in {\mathbb {Q}}^+$$pj∈Q+ and each machine has a speed $$s_i \in {\mathbb {Q}}^+$$si∈Q+. The goal of the graph balancing problem with speeds is to schedule each job j non-preemptively on one of a subset $${\mathcal {M}}_j$$Mj of at most 2 machines so that the makespan is minimized. This is a $$\textsf {NP}$$NP-hard special case of the makespan minimization problem on unrelated parallel machines, where for the latter a longstanding open problem is to find an approximation algorithm with approximation ratio better than 2. Our main contribution is an approximation algorithm for the graph balancing problem with two speeds and two job lengths with approximation ratio $$(\sqrt{65}+7)/8 \approx 1.88278$$(65+7)/8≈1.88278. In addition, we consider when every $${\mathcal {M}}_j$$Mj has no cardinality constraints, this is the restricted assignment problem in the uniform parallel machine setting. We present a simple $$(2-\alpha /\beta )$$(2-α/β)-approximation algorithm in this case when every job has one of two job lengths $$p_j \in \{\alpha , \beta \}$$pj∈{α,β} where $$\alpha